GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?
(1) The greatest common divisor of 2x and 2y is 2y
(2) The least common multiple of 2x and 2y is 20
Target question: What is the value of y? Given: The greatest common divisor of 3x and 3y is 6 This means that, if we examine the prime factorization of 3x and prime factorization of 3y, they will share exactly ONE 3 and ONE 2.
That is:
3x = (2)(3)(?)(?)(?)(?)
3y = (2)(3)(?)(?)(?)(?)
NOTE: Both prime factorizations might include other primes, BUT there is no additional overlap beyond the ONE 3 and ONE 2.
Notice that if we divide both sides of both prime factorizations by 3, we get:
x = (2)(?)(?)(?)(?)
y = (2)(?)(?)(?)(?)
This tells us that
the greatest common divisor (GCD) of x and y is 2.
Statement 1: The greatest common divisor of 2x and 2y is 2y We already know that...
x = (2)(?)(?)(?)(?)
y = (2)(?)(?)(?)(?)
So,
2x = (
2)(2)(?)(?)(?)(?)
2y = (
2)(2)(?)(?)(?)(?)
This tells us that the greatest common divisor (GCD) of 2x and 2y =(
2)(2) =
4.
The statement tells us that the GCD of 2x and 2y is 2y, which means 2y =
4Solve the equation to get
y = 2. PERFECT!!
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: The least common multiple of 2x and 2y is 20 There are several values of x and y that satisfy statement 2 (as well as satisfying the given information). Here are two:
Case a: x = 2 and y = 10. In this case, 2x = 4 and 2y = 20, and the least common multiple of 4 and 20 is 20. Also notice that 3x = 6 and 3y = 30, and the GCD of 6 and 30 is 6, which satisfies the given information. In this case
y = 10Case b: x = 10 and y = 2. In this case, 2x = 20 and 2y = 4, and the least common multiple of 20 and 4 is 20. Also notice that 3x = 30 and 3y = 6, and the GCD of 30 and 6 is 6, which satisfies the given information. In this case
y = 2Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer:
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